OEIS describes this sequence as a
bisection of the Fibonacci Sequence, listing the odd terms – but they include
both 1s in this sequence and one of these must be an even numbered term.I have elected to correct this in my data
listed below.

I want the inverse of my sequence
number to look like this digit sequence.So I then take this number to Wolfram Alpha (www.wolframalpha.com ) and find that it
is equal to

9.999970000029999990000000000000000000000000000000

000000...
× 10^11

I take the first 18 digits to form the
Sequence Number (because every digit after that is a zero):

999,997,000,002,999,999

Please notice that it looks very
similar to a sequence number used for Tribonacci sequences.The three parts are 999996, 1000002, and
999999 (the 6 in the first part and the 1 in the second part overlap and show
up as a 7 in the Sequence Number).These suggest it is a 3, -3, 1 Tribonacci Sequence Number.This means we can produce a list of the
Triangular Numbers the way they are defined, but we can also produce them as
the 3, -3, 1 Tribonacci Sequence.Both
are the same.

But we still have not tested it to see
if it really works.It’s time to do
that now.

Since this sequence is not included in
the OIES database, we can run a quick spreadsheet simulation to check it:

Non-Zero Term

The 3,-1,-1 Tribonacci
Sequence

0

0

1st

1

2nd

3

3rd

8

4th

20

5th

49

6th

119

7th

288

8th

696

9th

1,681

10th

4,059

11th

9,800

12th

23,660

13th

57,121

14th

137,903

15th

332,928

16th

803,760

17th

1,940,449

It all looks good until the 16th
non-zero term.This is because the 17th
term has seven digits so it carries over and adds one to the 16th
term.And the 18th term
carries over to the 17th term … and so forth.

The last group should be 999 + 1 (for
subtracting a(n-2)) and + 1 again for being the last group.This gives us 1,001.

The first group should be 999 – 2 (for
adding 2*a(n-1)) which gives us 997.

Combining these two parts, being
careful to carry the extra digit from the last group to the first group gives
us:

998,001

But we have seen this Sequence Number
before, right at the very beginning of our romp through Sequence Numbers, and
we already know that it produces a counting sequence from “000” to “997” in
three digit strings.

Who knew that when kids learn to count
they are actually learning how to do a special case of the Fibonacci
Sequence!

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About Me

This blog is about a special class of numbers that I call Sequence Numbers. I have been working on them for a few years,and just recently things came together. Phillip is helping me get this material posted.